Newspace parameters
Level: | \( N \) | \(=\) | \( 126 = 2 \cdot 3^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 7 \) |
Character orbit: | \([\chi]\) | \(=\) | 126.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(28.9868145361\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | 4.0.211968.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 30x^{2} + 207 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
Coefficient ring index: | \( 2^{7} \) |
Twist minimal: | no (minimal twist has level 14) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 30x^{2} + 207 \) :
\(\beta_{1}\) | \(=\) | \( ( 2\nu^{3} + 18\nu ) / 3 \) |
\(\beta_{2}\) | \(=\) | \( 2\nu^{3} + 34\nu \) |
\(\beta_{3}\) | \(=\) | \( ( 4\nu^{2} + 60 ) / 3 \) |
\(\nu\) | \(=\) | \( ( \beta_{2} - 3\beta_1 ) / 16 \) |
\(\nu^{2}\) | \(=\) | \( ( 3\beta_{3} - 60 ) / 4 \) |
\(\nu^{3}\) | \(=\) | \( ( -9\beta_{2} + 51\beta_1 ) / 16 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).
\(n\) | \(29\) | \(73\) |
\(\chi(n)\) | \(1\) | \(-1\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
55.1 |
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−5.65685 | 0 | 32.0000 | − | 98.3767i | 0 | 195.794 | − | 281.627i | −181.019 | 0 | 556.502i | |||||||||||||||||||||||||||
55.2 | −5.65685 | 0 | 32.0000 | 98.3767i | 0 | 195.794 | + | 281.627i | −181.019 | 0 | − | 556.502i | ||||||||||||||||||||||||||||
55.3 | 5.65685 | 0 | 32.0000 | − | 204.749i | 0 | −41.7939 | + | 340.444i | 181.019 | 0 | − | 1158.23i | |||||||||||||||||||||||||||
55.4 | 5.65685 | 0 | 32.0000 | 204.749i | 0 | −41.7939 | − | 340.444i | 181.019 | 0 | 1158.23i | |||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 126.7.c.a | 4 | |
3.b | odd | 2 | 1 | 14.7.b.a | ✓ | 4 | |
7.b | odd | 2 | 1 | inner | 126.7.c.a | 4 | |
12.b | even | 2 | 1 | 112.7.c.c | 4 | ||
15.d | odd | 2 | 1 | 350.7.b.a | 4 | ||
15.e | even | 4 | 2 | 350.7.d.a | 8 | ||
21.c | even | 2 | 1 | 14.7.b.a | ✓ | 4 | |
21.g | even | 6 | 2 | 98.7.d.b | 8 | ||
21.h | odd | 6 | 2 | 98.7.d.b | 8 | ||
24.f | even | 2 | 1 | 448.7.c.e | 4 | ||
24.h | odd | 2 | 1 | 448.7.c.h | 4 | ||
84.h | odd | 2 | 1 | 112.7.c.c | 4 | ||
105.g | even | 2 | 1 | 350.7.b.a | 4 | ||
105.k | odd | 4 | 2 | 350.7.d.a | 8 | ||
168.e | odd | 2 | 1 | 448.7.c.e | 4 | ||
168.i | even | 2 | 1 | 448.7.c.h | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
14.7.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
14.7.b.a | ✓ | 4 | 21.c | even | 2 | 1 | |
98.7.d.b | 8 | 21.g | even | 6 | 2 | ||
98.7.d.b | 8 | 21.h | odd | 6 | 2 | ||
112.7.c.c | 4 | 12.b | even | 2 | 1 | ||
112.7.c.c | 4 | 84.h | odd | 2 | 1 | ||
126.7.c.a | 4 | 1.a | even | 1 | 1 | trivial | |
126.7.c.a | 4 | 7.b | odd | 2 | 1 | inner | |
350.7.b.a | 4 | 15.d | odd | 2 | 1 | ||
350.7.b.a | 4 | 105.g | even | 2 | 1 | ||
350.7.d.a | 8 | 15.e | even | 4 | 2 | ||
350.7.d.a | 8 | 105.k | odd | 4 | 2 | ||
448.7.c.e | 4 | 24.f | even | 2 | 1 | ||
448.7.c.e | 4 | 168.e | odd | 2 | 1 | ||
448.7.c.h | 4 | 24.h | odd | 2 | 1 | ||
448.7.c.h | 4 | 168.i | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 51600T_{5}^{2} + 405720000 \)
acting on \(S_{7}^{\mathrm{new}}(126, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 32)^{2} \)
$3$
\( T^{4} \)
$5$
\( T^{4} + 51600 T^{2} + \cdots + 405720000 \)
$7$
\( T^{4} - 308 T^{3} + \cdots + 13841287201 \)
$11$
\( (T^{2} - 2220 T + 1227492)^{2} \)
$13$
\( T^{4} + 15367056 T^{2} + \cdots + 26266196601792 \)
$17$
\( T^{4} + \cdots + 126735731638272 \)
$19$
\( T^{4} + \cdots + 551809934313408 \)
$23$
\( (T^{2} + 20292 T + 100711044)^{2} \)
$29$
\( (T^{2} - 9132 T - 1357906716)^{2} \)
$31$
\( T^{4} + 2274932736 T^{2} + \cdots + 86\!\cdots\!12 \)
$37$
\( (T^{2} + 11596 T - 1847926364)^{2} \)
$41$
\( T^{4} + 9071224896 T^{2} + \cdots + 28\!\cdots\!32 \)
$43$
\( (T^{2} + 22348 T - 10521464924)^{2} \)
$47$
\( T^{4} + 20120477952 T^{2} + \cdots + 12\!\cdots\!88 \)
$53$
\( (T^{2} + 124308 T + 13614948)^{2} \)
$59$
\( T^{4} + 180594109200 T^{2} + \cdots + 78\!\cdots\!00 \)
$61$
\( T^{4} + 121774406928 T^{2} + \cdots + 82\!\cdots\!48 \)
$67$
\( (T^{2} + 217388 T + 11809058788)^{2} \)
$71$
\( (T^{2} - 225804 T - 95443772508)^{2} \)
$73$
\( T^{4} + 228009998400 T^{2} + \cdots + 26\!\cdots\!72 \)
$79$
\( (T^{2} - 1046452 T + 268612291876)^{2} \)
$83$
\( T^{4} + 478841902608 T^{2} + \cdots + 21\!\cdots\!08 \)
$89$
\( T^{4} + 258250641984 T^{2} + \cdots + 15\!\cdots\!52 \)
$97$
\( T^{4} + 42611214336 T^{2} + \cdots + 39\!\cdots\!12 \)
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